# Translations of Encyclopedia about Mathematics

## Theory of Games

The theory of games is concerned with strategic games whose result depends on the behaviour of the players concerned. It makes up a part of the theory of decision making and, as a game, refers to behaviours in society, politics or social classes. This allows us to simulate economic competition, the behaviour of the competition, conflicts and cooperation, and to evaluate them all.

This part of mathematics is also applied for business research (operation-research) and for economic theories.

An important characteristic of strategic games is that the adversary or a player is not passive but rather that it negotiates strategically. Each game proceeds according to set rules, as is the case within groups having certain structures. A common goal is to determine the optimal behaviour of one or more players.

There are three ways a player can decide:

by a conscious decision
by accepting risks
by accepting uncertainty

According to the procedure and type of the game, the players can make their moves all at once or one at a time.

Contrary to a normal game, where chance is involved (such as in a game of roulette or using dice), each player is able to influence the course and outcome of the game with their strategy. Examples of strategic games are chess or go, business negotiations, discussions for arms reduction, or divorce proceedings. Each "game" is defined according to previously set rules. As with chess, the number and possible moves are previously set.

The game strategy describes its plan, which is subordinate to the behaviour of each move, where each game move results in a certain outcome.

By rule, the decision of each player invokes a reaction by all the adversaries, where each move is considered a decision node in a decision tree.

If strategy A is noticeably more advantageous than strategy B, we refer to it as a dominant strategy, such as when interest rates of 3 to 6% apply to an account held at bank A while interest in accounts held at bank B are between 1.5 to 3%. In reality, game situations are more common and more complicated. When a divorcing couple is negotiating how to divide up their property or when the leaders of some states discuss peace or war, many factors are involved. In such strategic situations, diplomatic errors could completely change the strategy chosen by the various players and lead to possible avalanche effects.

So that we would not make decisions only according to common sense (which is often wrong), it is recommended to study the rule of game theory. A good computer chess program holds to the game rules, for which reason it usually wins.

Game trees are game situations in the form of decision nodes. With more complex games, graphical depiction has its limitations. For example, 10,120 nodes would be necessary for a game tree to explain chess. Using algorithms, each decision point or node can be calculated for the best possible outcome (minimax algorithm).

Games between two people are more simple. In games in general, each person is affected by the behaviour or decision made by each other player. In some games, one player may gain what another loses, in which case cooperation does not pay off.

Cooperation is an important issue in game theory because, in social settings, cooperation (contrary to chess) can bring greater profits to one or several cooperating players.

For example, there exists models of the form "you to me then I to you", where a player always reacts the same like its adversary who moved before them. If cooperation starts in such a game, then the occasional egoistic move will not deter tendencies towards cooperative behaviour.

The dilemma of the prisoners is a known example of a game where one does not benefit at the expense of another because, in this dilemma, when neither of the prisoners accuses the other to become the crown witness and receive the lower sentence, both of them receive a higher sentence.

In long term relations, profits from cooperation may invoke a tendency to greater cooperation because the cooperating players learn from the benefits and accommodate their strategies in light of this. The profits from such strategies are not yielded after each move but rather after a certain time.

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